Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.1 + 4 + 7 + ... + (3n - 2) = (3n - 1)

Question

What will be an ideal response?

Answer 1

First we show that the statement is true when n = 1.
 For n = 1, we get 1 = (3(1) - 1) = 1.
This is a true statement and Condition I is satisfied.

Next, we assume the statement holds for some k. That is,
  is true for some positive integer k.
We need to show that the statement holds for k + 1. That is, we need to show that
 
So we assume that  is true and add the next term, , to both sides of the equation.
1 + 4 + 7 + ... + (3k - 2) + 3(k + 1) - 2 = (3k - 1) + 3(k + 1) - 2
 = [k(3k - 1) + 6(k + 1) - 4]
 = (3k² - k + 6k + 6 - 4)
 = (3k² + 5k + 2)
 = (k + 1)(3k + 2)
 = (3k + 3 - 1)
 = (3(k + 1) - 1)
Condition II is satisfied. As a result, the statement is true for all natural numbers n.